SOLUTION: 1. Solve right triangle, ABC, giving the following information: Angle A=60 degrees, side c = 15 feet. Round angles to the nearest degrees, sides to nearest foot. 2. A ladd

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Question 356746: 1. Solve right triangle, ABC, giving the following information:
Angle A=60 degrees, side c = 15 feet. Round angles to the nearest degrees, sides to nearest foot.
2. A ladder rests against a house, making an angle of 52 degrees with the ground. If the ladder touches the house at a point 52 feet from the ground,, how long, to the nearest foot, is the ladder?
3. A boy whose eye level is two-feet above the ground is standing 18 fet from a tree. His angle of elevation to the top of the tree is 26 degrees. How high, to the nearest foot is the tree?
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Found 2 solutions by mananth, jsmallt9:
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
Angle A=60 degrees, side c = 15 feet. Round angles to the nearest degrees, sides to nearest foot.
this is a 30-60-90 right triangle
the sides are in the ratio of 1:2:sqrt3
Angle = 60 so angl c = 30 . if c = 15 ft a= 15sqrt3 & b = 30
2. A ladder rests against a house, making an angle of 52 degrees with the ground. If the ladder touches the house at a point 52 feet from the ground,, how long, to the nearest foot, is the ladder?
the ladder ,the wall and the ground distance of ladder form a right triangle.
Angle = 52 deg. distance on ground = 52 feet.
Cos angle = base /hypotenuse
cos 52 =52 /hypotenuse
hypotenuse = 52/cos 52
ladder length = 84 feet.
3. A boy whose eye level is two-feet above the ground is standing 18 fet from a tree. His angle of elevation to the top of the tree is 26 degrees. How high, to the nearest foot is the tree?
Tan 26 = height /base
Tan 26 * 18 = height
8.8 feet.
8.8 +2 = 10.8 height of tree
...

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Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
1. Solve right triangle, ABC, giving the following information:
Angle A=60 degrees, side c = 15 feet. Round angles to the nearest degrees, sides to nearest foot.
Since this is a right triangle and with angle A being 60 degrees, angles B and C must be 30 and 90 degrees. But which one is 30 and which one is 90? Either you have left something out or you have some naming convention for right triangles which puts the right angle in a fixed position in the name. Since I am not familiar with any such naming convention, I am going to solve the problem both ways:

With angle B as the right angle and side b the hypotenuse. This makes angle C the 30 degree angle. Using the 30-60-90 right triangle pattern we know that the hypotenuse is twice the side opposite 30. That makes side b = 30. And the side opposite the 60 degree angle is times the side opposite 30. This makes side a = .

With angle C as the right angle and side c the hypotenuse. This makes angle B the 30 degree angle. Using the 30-60-90 right triangle pattern we know that the hypotenuse is twice the side opposite 30. In other words, the side opposite the 30 degree angle is one-half the hypotenuse. That makes side b = 15/2. And the side opposite the 60 degree angle is times the side opposite 30. This makes side a = .

2. A ladder rests against a house, making an angle of 52 degrees with the ground. If the ladder touches the house at a point 52 feet from the ground,, how long, to the nearest foot, is the ladder?

It may help to draw a picture. The side of the house is vertical, the distance along the ground is horizontal (making the the angle a right angle) and the ladder is the hypotenuse. The angle of 52 degrees is between the ladder and the ground. The distance along the ground, 52 feet, is the side adjacent to the 52 degree angle. Since we want to find the length of the ladder, the hypotenuse, we need a Trig ratio which involves the adjacent side and the hypotenuse. This would be cos (adjacent/hypotenuse) or sec (hypotenuse/adjacent). Either can be used. The Algebra is simpler if you use the one that puts the unknown in the numerator. This would be sec in this problem. But calculators don't usually have sec buttons so finding a value for sec is not impossible but a little trickier than finding a value for cos. Rather than explaining how to find sec on a calculator, I am going to use cos:
cos(52) = 52/x
Multiplying both sides by x:
x*cos(52) = 52
Dividing both sides by cos(52):
x = 52/cos(52)
We can mow use our calculators to find the cos:
x = 52/0.6156614753256583 = 84.4620007651026853
To the nearest foot, the ladder is 84 feet long.

3. A boy whose eye level is two-feet above the ground is standing 18 fet from a tree. His angle of elevation to the top of the tree is 26 degrees. How high, to the nearest foot is the tree?

Again a picture might help. The boy is a vertical segment with his eyes 2 feet from the ground. From his eyes we will draw a right triangle:
What we will do to solve this is:
  1. Use Trig to find the vertical side of the triangle.
  2. Since the vertical side of the triangle is only from the spot 2 feet up to the top, we will need to add 2 to the vertical side of the triangle to get the full height of the tree.

Finding the vertical side. The vertical side is opposite to the 26 degree angle and the horizontal side is adjacent to it. So we want a Trig ratio that involves opposite and adjacent. So we will use tan (opposite/adjacent) or cot (adjacent/hypotenuse). This time the ratio with the unknown in the numerator is also the one on our calculators: tan!
tan(26) = x/18
Multiply both sides by 18:
18*tan(26) = x
Using our calculators on the tan:
18*0.4877325885658614 = x
8.7791865941855052 = x
Now we add the 2 (10.7791865941855052) and round to the nearest foot giving:
The height of the tree, to the nearest foot, is 11 feet.
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